What is it like to live in Tokyo from December to March?

>What is there to do in town and in the Tokyo vicinity?
Check its tourist website.
http://www.tourism.metro.tokyo.jp/english/

>What is the holiday/Christmas season like?
Christmas itself is not a holiday in Japan. So there is little difference from normal days.

>Are all the tourist attractions/museums/tours open/accessible?

Yes.

>How cold does it get in the winter?
Look at the average temperature.
http://gojapan.about.com/od/weathergeography/ig/Japan-Weather-Charts/Tokyo-Weather.htm

>Are things less expensive than the Spring and Fall?
No. The same.

>Is it a good time of year to take side trips to other parts of Japan like Hokkaido or Kyushu?
Hokkaido has a lot of snow. But you can visit just like you can visit in any other seasons.

>Are the nearby hot spring resorts near Tokyo open and in season?
Yes.

>Is it a good time to take a quick trip to Seoul, Shanghai or other Asian cities?
You can visit those cities any time in one year.

>What is the best kind of apartment setup to get in Tokyo for the winter?

It depends on what you want.